Problem 108 - Chemistry A Molecular Approach

Question

At what temperatures are the readings on the Fahrenheit and Celsius thermometers the same?

Step-by-Step Solution

Answer

\text{The temperature at which Fahrenheit and Celsius scales read the same is } -40 \text{ degrees.}

Step 1: Write down the temperature conversion formula

To find the temperature at which the Fahrenheit and Celsius scales read the same, use the conversion formula between Fahrenheit (\text{F}) and Celsius (\text{C}): \( \text{F} = \frac{9}{5} \text{C} + 32 \)

Step 2: Set the Fahrenheit temperature equal to the Celsius temperature

Since we're looking for the point where the Fahrenheit and Celsius temperatures are the same (\text{F} = \text{C}), replace \text{F} with \text{C} in the conversion formula: \( \text{C} = \frac{9}{5} \text{C} + 32 \)

Step 3: Rearrange the equation to solve for C

Subtract \(\frac{9}{5}\text{C}\) from both sides of the equation to isolate terms involving \text{C} on one side: \( \text{C} - \frac{9}{5} \text{C} = 32 \)

Step 4: Combine like terms and solve for C

Combine like terms by finding a common denominator, which in this case is 5, to get: \( \frac{5}{5} \text{C} - \frac{9}{5} \text{C} = 32 \), then combine the terms to get: \( -\frac{4}{5} \text{C} = 32 \). Finally, multiply both sides by the reciprocal of \(-\frac{4}{5}\) to solve for \text{C}: \( \text{C} = 32 \times \frac{-5}{4} \)

Step 5: Calculate the temperature

Perform the multiplication to find the temperature in degrees Celsius: \( \text{C} = -40 \) degrees Celsius, which is the same temperature in Fahrenheit.

Key Concepts

Fahrenheit to Celsius Conversion

Understanding how to convert temperatures from Fahrenheit to Celsius is essential for science and everyday situations alike, as different countries and scientific disciplines use various thermometric scales. The formula used for the conversion reflects a linear relationship between the two scales. In essence, the formula can be expressed as: \[\[\begin{align*} C = \frac{5}{9}(F - 32)\end{align*}\]\]When you want to convert a temperature from Fahrenheit to Celsius, you subtract 32 from the Fahrenheit temperature and then multiply by the fraction \(\frac{5}{9}\). For example, to convert 50 degrees Fahrenheit to Celsius:\[C = \frac{5}{9} (50 - 32) = \frac{5}{9} \times 18 = 10\]Therefore, 50 degrees Fahrenheit is equivalent to 10 degrees Celsius. This conversion is pivotal in the context of the original exercise which seeks the temperature at which readings on both Fahrenheit and Celsius thermometers are identical.

Thermodynamic Temperature Scales

Thermodynamic temperature scales, such as Fahrenheit and Celsius, are based on defined points, like the freezing and boiling points of water. Celsius is part of the metric system, where water freezes at 0 degrees and boils at 100 degrees, under standard atmospheric conditions.

Comparative Temperature Points

In contrast, the Fahrenheit scale sets the freezing point of water at 32 degrees and boiling at 212 degrees. This may seem less intuitive than Celsius, but it's widely used in the United States.

It is interesting to note that these scales intersect at a specific point. The exercise you encounter addresses this intersection, and the solution reveals that both Fahrenheit and Celsius scales read -40 degrees at the same moment. Understanding these scales and their relation can be incredibly helpful for interpreting weather forecasts, cooking instructions, and scientific data.

Solving Linear Equations

Solving linear equations is a foundational skill in algebra. The equation from the exercise is a classic example of a linear equation, and the solution involves isolating the variable C (Celsius). Linear equations can often be recognized by their standard form of can be expressed as: \[ax + b = 0\]where \(a\) and \(b\) are constants, and \(x\) is the variable we want to solve for.

Isolating the Variable

When we solve linear equations, we perform operations that simplify and isolate the variable. In the original exercise, this involved subtracting terms with C from both sides and then multiplying by the reciprocal of the coefficient in front of C. The goal is to have C alone on one side of the equation, allowing us to solve for the temperature at which both thermometer readings are identical.

Solving linear equations requires understanding and applying properties of equality and operations, which are skills used throughout all levels of mathematics.